Rinse and pat dry the leeks, then trim off the bottom and the dark green leaves at the top, leaving the light green and white part. Slice the leeks thin, then set aside.
Rinse and pat dry the asparagus, then cut off the woody bottoms of the asparagus, and chop the spears into thirds.
Bring a large frying pan to a medium heat and add in the olive oil and butter. Sauté the sliced leeks until translucent and soft, then add in the garlic, cook for an additional 2 minutes.
Add in the asparagus and cook until slightly softened. Sprinkle with salt and pepper, stir to combine. Turn off the heat; set aside.
Preheat the oven to 400º and line a sheet pan with parchment paper. Defrost the puff pastry dough on the counter at least 30-40 minutes so that it is a little more pliable. Add to a baking sheet with parchment paper.
Add the sautéed leeks and asparagus to the center of the pastry dough in an even layer, then top with shredded mozzarella cheese.
Brush the egg around the edges of the tart. Bake for 18-20 minutes or until the sides are golden brown.
Top with fresh herbs (thyme or rosemary), then let sit for at least 3-5 minutes before slicing and serving.
Notes
On a lightly floured surface, roll out the dough enough so that any creases and breaks mold together. This also makes the dough less sticky and pliable. You could also roll the dough between two pieces of parchment paper.Be sure to rinse the leeks really well after trimming and separating. The layers in between the leeks tend to hold a lot of dirt and residue.